“The real danger is not that computers will begin to think
like men, but that men will begin to think like computers.” (Sydney Harris)

The sprit of geometry versus the spirit of finesse Living as
we do in the so-called “information age”, it is perhaps not surprising that the
dominant model in contemporary psychology identifies thinking with information
processing or computation. According to this model, human thought is nothing
more than a set of rule-governed operations on various bits of information, that could in principle be performed by a
digital computer. The two main assumptions behind this model are: first, that the world consists of a collection of
information that can be clearly defined - usually in mathematical terms; and,
second, that knowledge is generated through the application of rules to this
information. Advocates of the model argue that it can explain all forms of intelligent
activity from deductive reasoning through common sense judgment to practical
know-how. Thus good judgment, which was once considered the hallmark of the
educated person, is now widely believed to have less value than technical
competence. Indeed, with the development of so-called “expert systems’ which
simulate the reasoning processes of experts in various disciplines, some believe
that we will eventually be able to replace fallible human judgment with the certainty
of mechanical calculation.

The aim of this article is to cast doubt on the information
processing model of thinking by reference to the ideas of the great seventeenth
century French philosopher, Blaise Pascal (1623-1662).
A genius of the first order, Pascal is credited, among other things, with inventing
the first digital computer - a geared machine that could add and subtract
numbers up to eight digits. But he would have had little sympathy with the
widely held belief that the mind is nothing more than a sophisticated computer.

The basis for Pascal s rejection of the computer model of the mind can be found
in a distinction he makes in his most famous work, the Pensées,
between the spirit of geometry and the spirit of finesse. This distinction is
designed to show that in addition to what we now call information processing,
the mind also functions at a more intuitive level. To explain the difference
between these two modes of thinking, we can say that while the spirit of
geometry analyses phenomena into clearly definable parts and uses deductive
reason to construct a system of knowledge based on rigorous proof, the spirit
of finesse concerns ideas and perceptions which cannot be precisely defined or
broken down into parts, and uses intuitive reason to make sense of the relevant
phenomena as a whole. Furthermore, while the spirit of geometry results in
conclusions that command universal assent, the spirit of finesse results in fallible
judgments about which intelligent people may from time to time disagree.

According to Pascal, while the geometrical mind is rigorous
and exact, it is also “slow, rigid and inflexible.” The intuitive mind, by
contrast, “has a suppleness of thought which fastens at once upon the various
pleasing qualities of what it loves.” Such a mind is “accustomed to judge at a
single glance”, and it does so “tacitly, naturally, and without technical
rules.” However, in distinguishing between these two ways of thinking, Pascal
makes it clear that both are important ingredients in a balanced intellectual
outlook. The trouble arises if we focus too exclusively on one or the other.
Thus, while mathematicians who are only mathematicians “do not at all
understand matters of feeling, seeking principles and being unable to see at a
glance,” “men of intuition who are only intuitive cannot have the patience to
reach to first principles.” While the ideal is a combination of the spirit of
geometry and the spirit of finesse, the tendency of the Western intellectual
tradition has been towards what might be called geometrical imperialism - that
is, the extension of geometrical thinking into every area of intellectual life
in the mistaken belief that it is the only possible approach to reality.

To counter geometrical imperialism, I wish briefly to draw
attention to the important role played by intuitive thinking in three different
subject areas: history, science, and mathematics.

History
History is a good example of a subject area in which intuitive understanding
rather than geometrical explanation is the appropriate intellectual mode. Since
history trades in concepts that resist precise mathematical specification, It
cannot be reduced to a measurable science, For example, an important factor in
determining the outcome of a battle might be the “morale’ of the troops; but “morale”
is an inherently vague, qualitative term; and while you may be able to measure
the troops’ height or weight, you cannot measure their morale. Certainly,
statistics have a role to play in history; but any attempt to reduce historical
explanations to numbers can only result in absurdity, as the following example from
the American historian Barbara Tuchman shows.

In a quantification study of the origins of World War I which I have seen, the
operators have divided all the diplomatic documents, messages, and utterances
of the July crisis into categories labeled ‘hostility’, ‘friendship’. ‘frustration’, ‘satisfaction” and so on, with each statement
rated for intensity on a scale from one to nine, including fractions. But no pre-established
categories could match all the private character traits and public pressures
variously operating on the nervous monarchs and ministers who were involved.
The massive effort that went into this study brought forth a mouse - the less
than startling conclusion that the likelihood of war increases in proportion to
the rise in hostility of the messages.

The absurdity of the above study is apparent from the fact
that it reaches conclusions that would have been readily apparent to a
historian of even average historical insight. In any case, even if it made
sense to organise the relevant historical documents
into categories and attach numbers to them, this would not obviate the need for
judgment; for, as Tuchman observes, the problem would remain of the choice of
categories - which are not ‘revealed doctrine’. In other words, judgment is
required to decide that the appropriate categories in terms of which to analyse
the data are ‘hostility” and “friendship” rather than, say, “insecurity” and
“envy”. Essentially, what we take the facts to be itself
depends on our pre-existing theories.

The motivation behind such quantification studies is to find
repeatable patterns in history but this remains an empty dream. For history is
by its very nature concerned, not with the abstract and general, but with the
concrete and particular. Moreover, the meaning and significance of particular
events cannot be understood in isolation, but only in the overall context in
which they take place. Since historical events are unique and unrepeatable, the
most for which we can hope when we inquire into, say, the cause of revolution
or the consequences of neutrality, are similarities rather than identities, and
tendencies rather than laws. Given that history is a matter of intuitive
understanding and judgment rather than geometrical explanation and proof, it is
not surprising that historians rarely achieve a consensus. However, this does
not vitiate history as a discipline, but simply reflects the nature of the
subject matter with which it deals.

Science
Despite the importance given to measurable quantities and law-like regularities
in traditional accounts of the scientific method, intuitive understanding also
plays a crucial role in scientific thinking. For, contrary to the traditional
inductive view of science, scientific theories cannot be derived mechanically from
the facts, but require a leap of creative insight. For example, despite the
enormous difference between the principles of Newtonian mechanics and those of
general relativity theory, they are, over a wide range consistent with the same
experimental facts. Einstein was not in possession of any facts that were not
available to physicists working in the Newtonian paradigm when he developed
relativity theory. He simply interpreted the existing facts differently, and
then went on to make testable predictions on the basis of them. This explains
his comment- which
could easily have been made by Pascal that ‘Laws are only reached by non-logical
methods. To make a law one has to have an intellectual love of the subject.’

What this suggests is that while scientific thinking within
a paradigm, or over-arching theory, might be geometric in nature, the
development of new paradigms is a matter of creative insight rather than
geometrical proof. Moreover, what we take the facts to be itself
depends on our pre-existing theories. Consider, for example, the statement: ‘Sulphur burns in air combining
with oxygen.’ While this might appear to be an unambiguous fact, the history of
science shows that it is actually a theory dependent observation. For prior to
the modern theory of combustion as developed by Lavoisier (1743-1794) at the end
of the 18th century, combustion was explained in terms of the giving off of a colourless, odorless and weightless substance called
phlogiston. So before Lavoisier, the relevant statement would have read. ‘Sulphur burns in air
releasing its phlogiston.’

The above discussion implies that, contrary to the traditional account of the
scientific method which says that theories are derived from neutral facts using
geometrical reason, it would be more accurate to say that facts are theory
dependent and that the theories on which they depend are the result of various
acts of creative insight. Reverting to Pascal’s terminology, we might conclude that
the spirit of finesse casts a deep shadow over the whole domain of science.

So why is the spirit of finesse not more readily apparent in
the sciences, and why is it so widely believed that theories can be
mechanically derived from the observable facts? Following the biologist, Peter
Medawar, I think that the reason derives from the way in which science is
written, For there is a sense in which the vast
majority of science writing is science fiction. I do not mean by this that the
results are false but rather that the report on the way in which they were
derived is false. Very often in science, the idea that grows into a testable
hypothesis comes before the evidence, and the scientist then looks to the
evidence to confirm her idea. However, when she comes to write up her work in a
science paper, the actual process of thought is reversed so that rather than admitting
that it was the idea that led to the selection of the data, she begins with the
data and implies that it was the evidence alone that led her to the conclusion.
What is wrong with this is that it covers up the creative nature of science in
which finesse plays a role and leaves us with the impression, not of an
open-ended and fallible process of enquiry, but of a completed and timeless
body of doctrine.

Mathematics While mathematics
is, of course, the homeland of geometrical explanations, it turns out that
intuitive understanding even has a role to play in this discipline. For as
Pascal observes, while we may use deduction to reason from axioms to theorems,
the axioms themselves are not subject to proof. All proof must end somewhere,
and it ends with first principles that we can justify only on the basis of intuitive
understanding. Furthermore, even when it comes to proving theorems, the French
mathematician Henri Poincare (1854 - 1912) claimed that great mathematicians
are guided as much by the intuition of beauty as by mechanical calculation:

Mathematical work is not a ample mechanical
work, and it could not be entrusted to any machine, whatever the degree of
perfection we suppose it to have been brought to. It is not merely a question
of applying certain rules, of manufacturing as many combinations as possible
according to certain fixed laws. The combinations so obtained would be
extremely numerous, useless and encumbering. The real work of the discoverer
consists in choosing between these combinations with a view to eliminating
those that are useless, or rather nut giving himself
the trouble of making them at all. The useful combinations are precisely the most
beautiful, I mean those that can most charm that special sensibility that all
mathematicians know, but of which laymen are so ignorant.

At a more esoteric level, Roger Penrose, in his book The
Emperor’s New Mind, contends that one of the implications of GodeI’s Incompleteness Theorem is that the way in which we
decide whether or not certain mathematical statements are true is intuitive,
rather than rule-governed. Be that as it may, we have all, at some level, had
the ‘A-ha!” experience of suddenly being able to see how to solve a
mathematical problem - and this is surely a faint echo of the higher level intuitions
of the great mathematicians.

Limitations of the Information processing model

These brief excursions into history, science and mathematics
all suggest that Pascal’s spirit of finesse, or what we have called intuitive
understanding, has a crucial role to play in intellectual activity. If we further
accept that intuitive understanding cannot be reduced to a rule-governed
activity, then it follows that the information processing model of thinking is
inadequate. Against this, it might be objected that when we learn a new skill,
such as poetry analysis, or essay writing, or, indeed, playing a musical instrument,
the activity in question is usually broken down into simple steps and we are
then given guidelines in the form of simple rules on how to proceed. This is
true enough; but once we acquire expertise in a particular area, we eventually go
beyond the rules that we have been taught. Thus, when we begin learning how to
write an essay, it is generally a good idea to follow some simple guidelines,
but we cannot reduce the work of a great writer to a set of mechanical rules - and
the oeuvre of a creative genius, such as Shakespeare, transcends any imaginable
rules. As I understand it, this is the point Pascal is making in the following
quotation:

True eloquence makes light of eloquence, true morality makes
light of morality: that is to say, the morality of the judgement, which has no rules,
makes light of the morality of the intellect. To make light of philosophy is to
be a true philosopher.

In its highest form, then, the spirit of finesse takes
flight and achieves a naturalness that lies entirely beyond rules. With
reference to the art of living, this is surely part of what we mean by wisdom.

Despite these comments, proponents of the information processing
model might insist that at some level our minds must always be following rules
and processing information even if we are not always aware of it. If intuition
is ‘the feeling of what the end must be without consciously going through every
step of the reasoning’ (Isaac Asimov), then even when we are not consciously
going through a sequential reasoning process, this must surely be happening at
the unconscious level. For otherwise intuitive insight and judgment become
entirely mysterious affairs. In response to this point, I would ask: But why
must this be happening? Why must our minds at some level be processing information?
Do we really want to claim that all areas of thoughtful activity, such as
scientific insight, writing poetry, and intellectual conversation, could in
principle be reduced to a set of rules that we are unconsciously following? Indeed,
I wonder if it even makes sense to speak of following a rule unconsciously.

In the absence of empirical evidence, I think there are two
reasons why many people are attracted by the information processing model.
First, we can all too easily become bewitched by our own rhetoric into taking a
metaphor ”The mind is a computer” - as a literal
truth. This seems to be a particular danger when we are dealing with the human
mind, which has frequently been understood in terms of the latest and most
fashionable technology. While it is doubtless illuminating to look at
similarities and differences between the mind, and a computer, there is no
particular reason to think that the mind is a computer.

The second possible reason for our prejudice in favour of
the information processing model concerns the nature of explanation itself.
Quite simply, we are so hooked into an atomistic and geometrical account of
reality, that we have no other conception of what it is to explain something.
Thus in the current Al debate, advocates of AI imply that the only alternative
to the mind-is-a-computer thesis is out-and-out mysticism. This, however, is a
false dichotomy. The way to steer a middle course between geometry and
mysticism is to accept that the spirit of finesse is an irreducible mode of
intellectual activity.

Perhaps the most fundamental weakness of the information
processing model of thinking is that it takes the idea of facts or information
for granted. Now, one would, of course, be a fool not to take note of the best
available facts when seeking to build up a picture of the world, and we are
likely to generate worthwhile ideas only if we have a good command of such facts.
At the same time, however, we must remember that facts do not simply rain down
from heaven. In his book, The Cult of Information Theodore Roszak
uses the word idea specifically to denote a master idea that informs an entire
way of thinking in a particular area. Using this terminology, Roszak argues that, “the mind thinks with ideas, not with
information.” Such ideas are not generated mechanically from the facts, but
rather inform what things show up as the facts in the first place, and how these
facts are interpreted. Indeed, as Roszak observes,
the belief that the world is a collection of facts, and the belief that the
mind is a computer are themselves ideas and not facts; and when we argue about
ideas, it is a question not of geometric proof, but of judgment and finesse.

Implications
What are the implications of the distinction between the spirit of geometry and
the spirit of finesse for us as educators in general, and TOK teachers in particular?
Let me mention three. First, we should want to avoid the view of education
espoused by Mr. Gradgrind in Charles Dickens’ novel,
Third Times:

‘Facts alone are wanted in life, Plant nothing else, and
root out everything else. You can only form the minds of reasoning animals upon
Facts; nothing else will ever be of any service to them.’

Contrary to Mr. Gradgrind, facts are clearly nor all
that is wanted in life; for education is not simply about conveying
information, but about teaching students how to reflect intelligently on
information, make meaningful connections, and come up with creative ideas. In
the current, sometimes mindless, rush to hook every classroom up to the
information superhighway, this is something that we would do well to remember.

Second, since intuition, judgment and creativity cannot be
reduced to a set of rules, we must accept the fact
that there can be no quick route to their acquisition, short of immersing
oneself in the subject-matter in question, and considering the relevant “facts”
from a variety of different perspectives. For the spirit of finesse is the
product not of a superficial survey and rote learning, but of wide reading and
personal reflection.

Third, while the information processing model implies that
every problem has a definite and clearly specifiable solution, I would argue
that not all thinking can be equated with problem-solving. While such an
equation may be appropriate in some areas, many important questions that
confront us – ‘What is the relationship between science and religion?’, ‘Do
animals have rights?’, ‘How should I live?’ etc. - do not have definite answers,
With such questions we should think in terms, not of problems that have a
clearly specifiable solution, but of difficulties that require thoughtful
illumination.

In conclusion, I think that in an increasingly computer
dominated and information-driven world, there is, as the opening quotation
suggested, a danger that people begin to think like computers and that they
come to value technical competence over wisdom, and mechanical calculation over
sound judgment. If we wish to avoid this danger and remain true to the
traditional ideal of the well-educated person, then, following Pascal, we must
seek to encourage in our students, not only the spirit of geometry, but also
the more elusive, yet equally valuable, spirit of finesse.